Sufficient conditions for linear long-wave instability of steady-state axisymmetric flows of an ideal liquid with a free boundary in an azimuthal magnetic field

The problem of linear stability of steady-state axisymmetric shear jet flows of an inviscid ideally conducting incompressible liquid with a free surface and “frozen-in” azimuthal magnetic field is analyzed. The sufficient conditions for theoretical (on semi-infinite time intervals) and practical (on finite time intervals) instability of these flows relative to small axisymmetric long-wave perturbations are obtained by the direct Lyapunov method. An a priori lower estimate indicating (at least) an exponential increase with time of small perturbations under investigation is constructed in the case when these conditions are valid for theoretical as well as practical instability. In addition, an illustrative analytic example of steady-state flows under investigation and small axisymmetric long-wave perturbations superimposed on them is constructed (according to our estimate, these perturbations increase with time).